Zagreb Connection Indices of Subdivision and Semi-Total Point Operations on Graphs

Tang, Jiang-Hua; Ali, Usman; Javaid, Muhammad; Shabbir, Khurram

doi:https://doi.org/10.1155/2019/9846913

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Abstract Introduction Preliminaries Applications and Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Volume 2019 | Article ID 9846913 | https://doi.org/10.1155/2019/9846913

Zagreb Connection Indices of Subdivision and Semi-Total Point Operations on Graphs

Jiang-Hua Tang,¹Usman Ali,²Muhammad Javaid,²and Khurram Shabbir³

Academic Editor: Juan L. G. Guirao

Received29 Oct 2019

Accepted09 Dec 2019

Published28 Dec 2019

Abstract

Representation or coding of the molecular graphs with the help of numerical numbers plays a vital role in the studies of physicochemical and structural properties of the chemical compounds that are involved in the molecular graphs. For the first time, the modified first Zagreb connection index appeared in the paper by Gutman and Trinajstic (1972) to compute total electron energy of the alternant hydrocarbons, but after that, for a long time, it has not been studied. Recently, Ali and Trinajstic (2018) restudied the first Zagreb connection index , the second Zagreb connection index , and the modified first Zagreb connection index to find entropy and acentric factor of the octane isomers. They also reported that the values provided by the International Academy of Mathematical Chemistry show better chemical capability of the Zagreb connection indices than the ordinary Zagreb indices. Assume that and denote the operations of subdivision and semitotal point, respectively. Then, the S-sum graphs are obtained by the cartesian product of and , where , are any connected graphs, and is a graph obtained after applying the operation S on . In this paper, we compute the Zagreb connection indices (, , and ) of the S-sum graphs in terms of various topological indices of their factor graphs. At the end, as an application of the computed results, the Zagreb connection indices of the S-sum graphs obtained by the particular classes of alkanes are also included.

1. Introduction

The importance of graph theory [1] is increased day by day in the various disciplines of science, especially in the field of chemistry and information science [2–5]. In view of Todeschini et al. [6], a topological invariant (TI) is the final outcome of a logical, systematic, and mathematical process that transforms features encoded in a molecular graph to a fixed real number. These TIs predict the chemical and physical features of the molecular graphs just on solubility, weight, connectivity, polarizability, surface tension, critical temperature, freezing, melting and boiling point, heat of evaporation, and heat of formation. A number of drugs, micromaterials, and crystalline materials which are strongly applied in several industrial networks and pharmaceutical fields are studied and evaluated on the basis or frame provided by the TIs (see [7–10]).

Moreover, TIs are presented into useful identity in the study of the quantitative structure activity relationships (QSARs) and quantitative structure property relationships (QSPRs), which connect the molecular graphs to their biological behaviour [11–16]. Mathematically, we represent this link as , where χ is the real-valued function, U is the molecular structure, and T is a real value that depends upon U. For more details, we refer to [17, 18, 19, 20]. The first TI was defined by Wiener in 1947 working on the boiling point of paraffin [21]. In 1972, Gutman and Trinajstić [22] defined degree-based TIs named as first Zagreb index and second Zagreb index. In the same paper, another TI is also defined, which got less tension of the researchers. Furtula and Gutman [23] reinvestigated this index with its various basic properties and called it by the name of forgotten index. Nowadays, many extended studies have been delivered on these invariants. For more study, we refer to [6, 24–27]. In brief, to date, many TIs with various properties have been defined (see [6, 27]). These TIs have been classified into various classes, but degree-based TIs are more popular than others (see [7–9, 28, 29]).

Also another TI is defined in the same paper [22] of Gutman and Trinajstić, but there is no more attention or study on this predictor for a long time. In 2018, Ali and Trinajstic [30] reinvestigated this topological index and called it modified first Zagreb connection index . They also reported (as per data provided by the International Academy of Mathematical Chemistry) that its chemical capability has more precise correlation coefficient values for the following thirteen physicochemical properties of octane isomers: density, heat capacity at temperature, standard enthalpy of vaporization, boiling point, heat capacity at pressure, enthalpy of vaporization, entropy, acentric factor, molar volume, total surface area, enthalpy of formation, standard enthalpy of formation, and octanol water partition coefficient. Du et al. [31] determined extremal alkanes by using modified first Zagreb connection index. In the same year, Ducoffe et al. [32] also characterized extremal graphs with respect to the modified first Zagreb connection index for trees and unicyclic graphs with or without a fixed girth. Shao et al. [33] also used the same molecular descriptor to find the extremal graphs in the class of alkanes and cycloalkanes under certain conditions.

In the computational graph theory, the operations on graphs played an important role in the studies of graph invariants in the complex structures into their factors. For the purpose, Yan et al. [34] defined the operations of subdivision and semitotal point on graphs and obtained the Wiener index of the resultant graphs. Eliasi and Taeri [35] used these operations to construct new families of S-sum graphs. Deng and Akhter [36, 37] computed the first Zagreb index , second Zagreb index , and forgotten index of the S-sum graphs. Liu et al. [38] derived the exact formulas for the first general Zagreb index on the S-sum graphs, where α is a real number. For more details about TIs and their applications, we refer to [39–48] and the references therein.

In this paper, we extend this study and compute Zagreb connection indices such as first Zagreb connection index , second Zagreb connection index , and modified first Zagreb connection index of the S-sum graphs , which are obtained by the cartesian product of and , where , are any connected graphs, and is a graph obtained after applying the operations S on . The rest of the paper is organized as follows: Section 2 covers the preliminary definitions of the Zagreb indices and Zagreb connection indices. Section 3 presents the main results, and Section 4 includes the conclusion.

2. Preliminaries

Let be a simple and connected graph with vertex set and edge set such that their cardinalities are called order and size of Q, respectively. Todesehini et al. [6] defined the generalized form of the degrees of vertices of the graph Q as , where . For k = 1 and k = 2, and are called the degree and connection number of the vertex b in the graph Q. In chemical modeling of molecular descriptors [49], chemical terms atom and bond are equal to graphical terms vertex and edge, respectively. For further basic terminologies of graphs, we refer to [50].

Definition 1. For a graph Q, the first Zagreb index , second Zagreb index , and forgotten index are defined asThese degree-based indices are defined by Gutman, Trinajstic, Ruscic, and Furtula (see [22, 23, 51]). These are frequently used to predict better outcomes in molecular structures such as entropy, heat capacity, ZE isomerism, acentric factor, and absolute value of correlation coefficient [52]. These indices also played an important role in the study of QSPR and QSAR (see [18, 19, 49]).
Corresponding to these degree-based TIs, the connection-based TIs are defined in Definition 2. For further studies of connection-based TIs, see [30, 53].

Definition 2. For a graph Q, the first Zagreb connection index , second Zagreb connection index , and modified first Zagreb connection index are defined asNow, we describe the subdivision and semitotal point operations on graphs as follows: (a) is a graph that is obtained by including a new vertex between each edge of the graph Q; (b) is a graph that is obtained from by adding the edges between old vertices which are adjacent in Q. More details of these operations can be found in [34, 35, 54, 55]. For further understanding, see Figure 1. The concept of S-sum graphs is defined as in Definition 3.

Definition 3. For and , assume that are connected graphs and the graph with vertex set and edge set is obtained after applying the operation S on . Then, the graphs with vertex set and edge set are called S-sum graphs if for either in and or in and , where , For more clearance, see Figures 2 and 3.
Now, we present some important results which are used in the main results.

Lemma 1 (see [56]). Let Q be a connected graph with n vertices and m edges. Then, , where equality holds if and only if Q is a free graph.

Lemma 2. Let Q be a connected and free graph with n vertices and m edges. Then, .

Proof. Proof is obvious using Hand-Shaking Lemma and Lemma 1.

3. Main Results

This section contains the main results of the first Zagreb connection index , second Zagreb connection index , and modified first Zagreb connection index on the S-sum graphs under the operations of subdivision and semitotal point.

Theorem 1. Let and be two connected and free graphs. Then, the first Zagreb connection indices of their S-sum graphs under the operations of subdivision and semitotal point are

Proof. (a) Let be a connection number of the vertex in the graph . Then,Consider By Lemma 2, we haveSince we haveBy Definition 1, we haveFinally, we obtain

Proof. (b) Let be a connection number of the vertex in the graph . Then,ConsiderUsing Definition 2 and Lemma 2, we haveAlso, considerFinally, we obtain

Theorem 2. Let and be two connected and free graphs. Then, the second Zagreb connection indices of their S-sum graphs under the operations of subdivision and semitotal point are

Proof. (a) Let be a connection number of the vertex in the graph . Then,ConsiderAlso, considerTherefore,

proof. (b) Let be a connection number of the vertex in the graph . Then,ConsiderAlso, considerConsequently,

Theorem 3. Let and be two connected and free graphs. Then, the modified first Zagreb connection indices of their S-sum graphs under the operations of subdivision and semitotal point are

Proof. (a) Let be a connection number of the vertex in the graph . Then,ConsiderAlso, considerFinally, we obtain

proof. (b) Let be a connection number of the vertex in the graph . Then,ConsiderAlso, considerConsequently,

4. Applications and Conclusion

Let and be two particular alkanes called by paths, where . Then, the Zagreb connection indices of their S-sum graphs as the consequences of the obtained main results are as follows:(1)(a) (b)(2)(a) (b)(3)(a) (b)

In this paper, we obtained the S-sum graphs using the operations of subdivision, semitotal point and cartesian product on two connected graphs in a certain order. Finally, we computed the first Zagreb connection index, second Zagreb connection index, and modified first Zagreb connection index of the obtained S-sum graphs in the terms of Zagreb indices and Zagreb connection indices of their factor graphs. Our results are interesting for pharmaceuticals [57–61]. Also, the problem is still open to compute the Zagreb connection indices for other families of graphs obtained by various operations on graphs.

Data Availability

The data used to support the findings of this study are cited at relevant places within the text as references.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this paper.

Acknowledgments

This work was jointly supported by the Natural Science Fund Project of Anhui Xinhua University (Grant no. 2017zr011).

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